The path integral formalism gives a very illustrative and intuitive understanding of quantum mechanics but due to its difficult sum over phases one usually prefers Schrödinger's approach. We will show that it is possible to calculate simple quantum phenomena by performing Feynman's sum over all paths staying entirely in real time. Once the propagator is obtained it is particularly easy to get the energy spectrum or the evolution of any wavefunction.
University of Michigan（密歇根大学）；RIKEN Cluster for Pioneering Research（理化学研究所先驱研究集群）；RIKEN Center for Quantum Computing（理化学研究所量子计算中心）；The University of Michigan（密歇根大学）
Feynman path integrals provide an elegant, classically inspired representation for the quantum propagator and the quantum dynamics, through summing over a huge manifold of all possible paths. From computational and simulational perspectives, the ergodic tracking of the whole path manifold is a hard problem. Machine learning can help, in an efficient manner, to identify the relevant subspace and the intrinsic structure residing at a small fraction of the vast path manifold. In this work, we propose the Feynman path generator for quantum mechanical systems, which efficiently generates Feynman pa…
Feynman's sum-over-histories formulation of quantum mechanics has been considered a useful calculational tool in which virtual Feynman histories entering into a coherent quantum superposition cannot be individually measured. Here we show that sequential weak values, inferred by consecutive weak measurements of projectors, allow direct experimental probing of individual virtual Feynman histories thereby revealing the exact nature of quantum interference of coherently superposed histories. Because the total sum of sequential weak values of multi-time projection operators for a complete set of or…
Perturbative quantum field theory usually uses second quantisation and Feynman diagrams. The worldline formalism provides an alternative approach based on first quantised particle path integrals, similar in spirit to string perturbation theory. Here we review the history, main features and present applications of the formalism. Our emphasis is on recent developments such as the path integral representation of open fermion lines, the description of colour using auxiliary worldline fields, incorporation of higher spin, and extension of the formalism to non-commutative space.
The Feynman path integral does not allow a "one real path" interpretation, because amplitudes contribute to probabilities in a non-separable manner. The opposite extreme, "all paths happen", is not a useful or informative account. In this paper it is shown that an intermediate parsing of the path integral, into realistic non-interfering possibilities, is always available. Each realistic possibility formally corresponds to numerous particle paths, but is arguably best interpreted as a spacetime-valued field. Notably, one actual field history can always be said to occur, although it will general…
The Yukawa interaction is considered in 0+1 dimensions as a pedagogical example to illustrate quantum field theory methods. From the quantum mechanical point of view the system is trivially exactly solvable, but this can be difficult to see from the path integral perspective. It is shown how despite initial appearances the perturbation series in terms of Feynman diagrams is consistent with the quantum mechanical picture for finite temperature.
In quantum statistical mechanics, Moyal's equation governs the time evolution of Wigner functions and of more general Weyl symbols that represent the density matrix of arbitrary mixed states. A formal solution to Moyal's equation is given by Marinov's path integral. In this paper we demonstrate that this path integral can be regarded as the natural link between several conceptual, geometric, and dynamical issues in quantum mechanics. A unifying perspective is achieved by highlighting the pivotal role which the response field, one of the integration variables in Marinov's integral, plays for pu…
Systems of interacting classical harmonic oscillators have received considerable attention in the last years as analog models for describing electromagnetically induced transparency (EIT) and associated phenomena. We review these models and investigate their validity for a variety of physical systems using two- and three-coupled harmonic oscillators. From the simplest EIT-$Λ$ configuration and two-coupled single cavity modes we show that each atomic dipole-allowed transition and a single cavity mode can be represented by a damped harmonic oscillator. Thus, we have established a one-to-one corr…
Our article [arXiv:2111.13727(2021)] argues that the phenomenology of interference that is traditionally regarded as problematic does not, in fact, capture the essence of quantum theory -- contrary to the claims of Feynman and many others. It does so by demonstrating the existence of a physical theory, which we term the "toy field theory", that reproduces this phenomenology but which does not sacrifice the classical worldview. In their Comment [arXiv:2204.01768(2022)], Hance and Hossenfelder dispute our claim. Correcting mistaken claims found therein and responding to their criticisms provides…
By investigating the Feynman Path Integral we prove that elementary quantum particle dynamics are directly associated to single compact (cyclic) world-line parameters, playing the role of the particles' internal clock, implicit in ordinary undulatory mechanics and indirectly observed for instance in Time Crystals. This allows us to formulate a novel purely four-dimensional stringy description of elementary particles as possible physics beyond quantum mechanics. The novelty of this approach is that quantum mechanics originates from a non-trivial compact nature of the minkowskian space-time. Our…
Standard Quantum Mechanics, although successful in terms of calculating and predicting results, is inherently difficult to understand and can suffer from misinterpretation. Entropic Dynamics is an epistemic approach to quantum mechanics based on logical inference. It incorporates the probabilities that naturally arise in situations where there is missing information. The development of Entropic Dynamics involves describing a particle in terms of a probability density, and then following the time evolution of the probability density based on diffusion-like motion and maximization of entropy. He…
Simulating quantum many-body systems on a classical computer generally requires a computational cost that grows exponential with the number of particles. This computational complexity has been the main obstacle to understanding various fundamental emergent phenomena in condensed matters such as high-Tc superconductivity and the fractional quantum-Hall effect. The difficulty arises because even the simplest models that are proposed to capture those phenomena cannot be simulated on a classical computer. Recognizing this problem in 1981, Richard Feynman envisioned a quantum simulator, an entirely…
We consider a quantum mechanical model for the high-order harmonic generation in bulk solids. The bandgap is assumed to be considerably larger than the exciting photon energy. Using dipole approximation, the dynamical equations for different initial Bloch states are decoupled in the velocity gauge. Although there is no quantum mechanical interference between the time evolution of different initial states, the complete harmonic radiation results from the interference of fields emitted by all the initial (valence band) states. In particular, this interference is shown to be responsible for the s…
The diagonal representation and optical equivalence theorem are the E. C. G. Sudarshan's mid 20th century adventures in non-classical optics. It basically deals with a quantum mechanical description of photons to explain the quantum properties of light. Inspired by Sudarshan's pioneering work we try to explain the every minute mathematical details of his paper "Equivalence of semi-classical and quantum mechanical descriptions of statistical light beams". In this article we are going to go through some of the basics in developing quantum optics, then land up in E.C.G's original work and try to …
We point out that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack in their standard form important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable (i.e. in multiply-connected spaces), and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, eith…
Classical physics fails where quantum physics prevails. This common understanding applies to quantum phenomena that are acknowledged to be beyond the reach of classical physics. Here, we make an attempt at weakening this solid belief that classical physics is unfit to explain the quantum world. The trial run is the quantization of the free radiation field that will be addressed by following a strategy that is free from operators or quantum-mechanical concepts
Quantum coupling between mechanical oscillators and atomic gases generating entanglement has been recently experimentally demonstrated using their subsequent interaction with light. The next step is to build a hybrid atom-mechanical quantum gate showing bosonic interference effects of single quanta in the atoms and oscillators. We propose an experimental test of Hong-Ou-Mandel interference between single phononic excitation and single collective excitation of atoms using the optical connection between them. A single optical pulse is sufficient to build a hybrid quantum-nondemolition gate to ob…
Coherent control of a quantum mechanical two-level system is at the heart of magnetic resonance imaging, quantum information processing, and quantum optics. Among the most prominent phenomena in quantum coherent control are Rabi oscillations, Ramsey fringes and Hahn echoes. We demonstrate that these phenomena can be derived classically by use of a simple coupled harmonic oscillator model. The classical problem can be cast in a form that is formally equivalent to the quantum mechanical Bloch equations with the exception that the longitudinal and the transverse relaxation times ($T_1$ and $T_2$)…
Observing quantum phenomena in macroscopic objects, and the potential discovery of a fundamental limit in the applicability of quantum mechanics, has been a central topic of modern experimental physics. Highly coherent and heavy micro-mechanical oscillators controlled by superconducting circuits are a promising system for this task. Here, we focus in particular on the electrostatic coupling of motion to a weakly anharmonic circuit, namely the transmon qubit. In the case of a megahertz mechanical oscillator coupled to a gigahertz transmon, we explain the difficulties in bridging the large elect…
在此之前，Bennet和Feynman问海森堡的不确定原理是否对量子计算机有限制（Quantum Mechanical Computers，Richard P. Feynman，Foundations of Physics，Vol. 16，No. 6，p597-531，1986）。费曼的回答是否定的。在本文中，我们将重新讨论离散时间傅里叶变换的不确定原理的相同问题。我们将表明，离散时间傅立叶变换不确定原理起着根本性的作用，表明肖尔的量子算法的类型具有有效的运行时间，并得出结论，离散时间不确定原理是一个援助，在我们目前的制定和理解肖尔的量子算法的类型。事实证明，对于这些算法，测量某个集合$T$中的元素的概率（在算法结束时）可以用有限傅立叶分析的时间限制和频带限制算子来表示。与这些算子相联系的是有限傅里叶变换不确定原理。不确定原理为上述概率提供了一个下限。我们将推导出这些类型的概率的一般下界。我们将这些下界称为量子算法不确定原理或QAUP。QAUP之所以重要，是因为它们让我们对衡量某些可取之处的概率有了一些了解。我们将使用这些下界来推导Shor的因子分解和离散对数算法。
A quantum system can be driven by either sinusoidal, rectangular, or noisy signals. In the literature, these regimes are referred to as Landau-Zener-Stuckelberg-Majorana (LZSM) interferometry, latching modulation, and motional averaging, respectively. We demonstrate that these pronounced and interesting effects are also inherent in the dynamics of classical two-state systems. We discuss how such classical systems are realized using either mechanical, electrical, or optical resonators. In addition to the fundamental interest of such dynamical phenomena linking classical and quantum physics, we …
